Solving a 20-Year Puzzle: A Sports Car Enthusiast's Homework

[undrcvrbro]

Homework Statement

A dedicated sports car enthusiast polishes the inside and outside surfaces of a hubcap that is a section of a sphere. When he looks into one side of the hubcap, he sees an image of his face 34.0 cm in back of the hubcap. He then turns the hubcap over, keeping it the same distance from his face. He now sees an image of his face 9.4 cm in back of the hubcap.

How far is his face from the hubcap?

Homework Equations

(1/f) = (1/p)+(1/q)
m= -q/p

The Attempt at a Solution

Well, I recognize that there is some sort of theoretical way of solving this, but really...im drawing a blank. My AP physics teacher is a little goofy, and when someone asked him for help on the same problem today we(his students) realized that the answer to the problem he had written down for 20 years was incorrect. Please, anyone...I need your help.

 

[Shooting Star]

If one is a convex mirror, the other side is concave. Given the data, can we tell which image is for which type? The dist of the source is the same in both cases. Is there anything else common for both?

 

[Moetasim]

I understand the frustration of encountering a problem that seems to have no clear solution. However, it is important to approach this problem with a scientific mindset and use the tools and equations at our disposal to find a solution.

First, let's analyze the given information. We know that the hubcap is a section of a sphere, and when the enthusiast looks into one side of the hubcap, he sees an image of his face 34.0 cm behind it. This means that the distance between the hubcap and his face (p) is 34.0 cm.

Next, we are told that when the hubcap is turned over, the image of his face is now 9.4 cm behind the hubcap. Using the equation (1/f) = (1/p)+(1/q), we can set up the following equation:

(1/34.0) = (1/34.0)+(1/q)

Solving for q, we get q = 34.0 cm. This means that the distance between the hubcap and his face when it is turned over is also 34.0 cm.

Now, using the equation m= -q/p, we can calculate the magnification of the image when the hubcap is turned over:

m = -(34.0 cm)/(34.0 cm) = -1

This means that the image is the same size as the object (his face) when the hubcap is turned over.

Finally, to find the distance between his face and the hubcap, we can use the equation m= -q/p again, but this time solving for p:

m = -1 = -(34.0 cm)/(p)

Solving for p, we get p = 34.0 cm.

Therefore, the distance between the enthusiast's face and the hubcap is 34.0 cm. This means that the hubcap is acting as a virtual image and is located at a distance of 34.0 cm from the enthusiast's face. I hope this helps solve the 20-year puzzle and provides a clear solution to the problem.